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Local surgery formulas for quantum invariants and the Arf invariant. (English) Zbl 1087.57006

Gordon, Cameron (ed.) et al., Proceedings of the Casson Fest. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences, Fayetteville, AR, USA, April 10–12, 2003 and the conference on the topology of manifolds of dimensions 3 and 4, Austin, TX, USA, May 19–21, 2003. Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 213-233 (2004).
The authors consider the problem of computing the Reshetikhin-Turaev-Witten quantum \(SU(2)\)-invariants at the fourth root of unity, \(\tau_4(M)\), for all closed oriented \(3\)-manifolds \(M\). They show in Section 2, Theorem 2.1, that this problem is \(\mathcal{N}\mathcal{P}\)-hard in the sense of complexity theory, that is, a polynomial time algorithm would yield polynomial time algorithms for all problems whose answers can be checked in polynomial time (the so called \(\mathcal{N}\mathcal{P}\)-problems). It is also remarked that a polynomial time algorithm exists for the class of \(3\)-manifolds that can be obtained by surgery on an integrally framed link whose Milnor’s \(\bar{\mu}\)-invariants of order \(\leq 3\) vanish.
The main tool, explained in Section 1, is a new formula for the Arf invariant \(\alpha(C)\) of a link \(C = \coprod_i C_i\) with even pairwise linking numbers : \[ \alpha(C) = \sum_i \alpha(C_i) + \sum_{i<j} \frac{1}{4} (\lambda(C_i,C_j) + {\text{lk}}(C_i,C_j)) + \sum_{i<j<k} \tau(C_i,C_j,C_k) \pmod 2 \] where \(\lambda\) is the unoriented Sato-Levine invariant and \(\tau\) is Milnor triple point invariant. This formula is proved in Section 4.2, after an exposition of the Brown invariant in Sections 3 and 4.1. It is local, as it involves only sublinks of \(C\) with \(3\) or fewer components, and depends only on linking numbers of curves near \(\partial F \cup S\) and on the Brown invariant of quadratic forms on \(H_1(F)\), where \(F\) is an immersed surface in \(S^3\) bounded by \(C\) whose singularities \(S\) are away from \(\partial F\).
The idea for the proof of Theorem 2.1 is to construct a class of \(3\)-manifolds \(M_c\) indexed by cubic forms \(c: (\mathbb{Z}/2)^n \rightarrow \mathbb{Z}/2\) over \(\mathbb{Z}/2\), such that \[ \tau_4(M_c) = \sum_{x \in (\mathbb{Z}/2)^n} (-1)^{c(x)}. \] For that the authors use the formula above for \(\alpha\) (corresponding to the form \(c\)), together with a formula previously obtained in their paper [Invent. Math. 105, 473–545 (1991; Zbl 0745.57006)], expressing \(\tau_4(M_c)\) in terms of Rokhlin’s invariant of the spin structures on \(M_c\), which is well-known to be related to the Arf invariant of characteristic sublinks in a surgery presentation of \(M_c\). Then \(\tau_4(M_c)\) just counts the number of zeros of the form \(c\), a well-known \(\mathcal{N}\mathcal{P}\)-hard computational problem.
For the entire collection see [Zbl 1066.57002].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)

Citations:

Zbl 0745.57006
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